The supposed proof that P != NP and your thoughts on mathematical advances

A week and a half ago on 8-Aug, a draft paper by Vinay Deolalikar, a well respected Principal Research Scientist at HP Labs, was released that supposedly proves that polynomial time (P) does not equal nondeterministic polynomial time (NP). To those unfamiliar with the P = NP problem, it’s hard to formally state in an easy to understand manner (most definitions involve Turing Machines), but the basic premise is this: For problems in which a computer can rapidly determine the correctness of a given solution, can computers also rapidly produce solutions to them? Problems that computers can rapidly verify solutions for are NP problems, problems that computers can rapidly solve are P problems, and the hardest to solve problems in the class of NP problems are called NP Complete. If a P solution can be found for an NP Complete problem, then P = NP.

An answer to whether or not P = NP has been researched for decades and has enormous implications for all aspects of modern life. Further, the Clay Mathematics Institute included P = NP as one of their seven one million dollar Millennium Prize problems. In a 2002 poll conducted by William Gasarch and published in ACM SIGACT News, only 9 theorists out of the 100 who responded stated that they believe P = NP, so it would be safe to say that the general consensus is that P != NP.

Integer Programming is an example of an NP Complete problem. I’m hopeful that eventually someone will come up with rapidly solving algorithm for IP in the same way that Dantzing’s Simplex algorithm revolutionized linear programming. Simplex isn’t a P solution - you can make bad pivot choices – but there are modern LP algorithms that are P solutions. (I remember spending hours as an undergraduate doing simplex pivots and kind of wonder if it’s still taught.) Now, obviously, I believe that P = NP, and I’m excited by the idea. A P solution for IP would be, IMHO, the biggest advance in mathematics since the invention of calculus, and it would revolutionize everything from transportation to genetic analysis. (Most problems can be restated as IP problems.) It would also be the end of cryptography as we know it, but major technological advances often have some pain associated with them.

Deolalikar’s 102 page draft proof set off a serious firestorm in the community of computation complexity researchers. A number of highly regarded theorists, including a couple of Fields Medal winners, have serious doubts about its correctness, and several potentially fatal flaws have been identified in it. Deolalikar has stated that he has been able to address the issues raised and that a revised version of the proof, weighing in at 126 pages, will be submitted to a journal for peer review this week. Regardless of whether or not the proof is correct, I hope Dr. Deolalikar’s paper is published. It’s an interesting (though extraordinarily complex) approach to a difficult problem, provides many new avenues for further research, and certainly seems to have people excited about the subject.

This brings me to my question. What do you expect to be the biggest mathematics or computation advance that will occur in your lifetime? What do you hope will be advanced in your lifetime? My answer is a solution for IP, but I’m curious to hear the thoughts of others.

Reference: Much of the debate over Deolalikar’s paper broke out on the blog of my favorite mathematics writer, Prof. Richard Lipton. You can read more at http://rjlipton.wordpress.com/